Insulating moiré homobilayers lack a threefold symmetric second-harmonic generation

Atoms within moir\'e bilayers relax in plane to minimize elastic energy [e.g., Cazeaux et al., J. Elast. 154, 443 (2023)]; such relaxation brings their space group symmetries down to P1. Here, the ab initio second harmonic generation (SHG) of twisted and atomistically optimized hBN bilayers was determined at four twist angles ($\ensuremath{\theta}=38.{21}^{\ensuremath{\circ}}, 60.{00}^{\ensuremath{\circ}}, 73.{17}^{\ensuremath{\circ}}$, and $98.{21}^{\ensuremath{\circ}}$) and for three displacements $\mathbit{\ensuremath{\tau}}$ measured away from the ground state $A{A}^{\ensuremath{'}}$ configuration. All moir\'e bilayers have a P1 space symmetry after structural optimization. This situation is quite different to monolayers with hexagonal lattices, which retain a threefold symmetry. We point out that the actual symmetries of the SHG reported for hBN bilayers on two experimental works do not coincide with the sixfold symmetric theoretical profiles they provide [either ${sin}^{2}(3\ensuremath{\phi})$ or ${cos}^{2}(3\ensuremath{\phi})$], and show that the intrinsic low structural symmetry of (atomically optimized) hBN bilayer moir\'es can in fact be read out from experimental SHG intensity profiles---which are tunable by $\ensuremath{\theta}$ and by the frequency of light $\ensuremath{\omega}$: The SHG is most definitely not sixfold symmetric because moir\'es do not retain a threefold symmetry. Furthermore, an extrinsic twofold symmetry of the SHG emission is realized by tilting the pump by an angle $\ensuremath{\alpha}$ away from the 2D material's normal, regardless of $\ensuremath{\theta}$ and $\ensuremath{\omega}$. The design of in-plane and ultrathin sources of SHG with low symmetry could be useful for the eventual creation of entanglement sources from 2D materials.